Find the ordered pair $(m,n)$, where $m,n$ are positive integers satisfying the following equation:
$$14mn = 55 - 7m - 2n$$
Answer: Looking at the form of the equation, we see that we have two linear terms and their product.  We thus apply Simon's Favorite Factoring Trick.  The given equation rearranges to $14mn + 7m +2n +1 = 56$, which can be factored to $(7m + 1)(2n +1) = 56 = 2\cdot 2\cdot 2\cdot 7$.  Since $n$ is a positive integer, we see that $2n +1 > 1$ is odd.  Examining the factors on the right side, we see we must have $2n + 1 = 7$, implying $7m+1 = 2^3$.  Solving, we find that $(m,n) = \boxed{(1,3)}$.